The "old" definition of spin liquid is a state that breaks no symmetry, e.g. the ground state is like a fluctuating liquid of spins. This is still very useful intuition for designing models and in experiments. Nevertheless, it is a fundamentally bad definition since it defines a spin liquid by what it isn't and misses the key physics.
The key physics of the spin liquid is the presence of long-range entanglement. This is the "new" definition of a spin liquid. Long range entanglement typically manifests itself in terms of emergent gauge fields, fractionalization of quantum numbers, emergent anyons, etc.
To understand what this means, take two examples.
First take a conventional SDW, say an AF state in the square lattice Heisenberg model. Break all symmetries by imposing external fields. Then the system will be gapped and can be smoothly deformed into a product state. Hence without symmetry the system is short-range entangled (can be deformed smoothly to a product state).
On the other hand, what we really mean by a spin liquid is a long range entangled state. Take such a spin liquid and let's assume its gapped as in your example. Break all symmetries including discrete symmetries. The state cannot be deformed into a product state without encountering a phase transition.
So to summarize, a spin liquid, by which I mean a long-range entangled state, can always have some magnetic order or other broken symmetry sitting on top of it. This symmetry breaking can be spontaneous, or if the Hamiltonian lacks the relevant symmetry, such magnetic order will be present on symmetry grounds. However, as long as you have long-range entanglement the physics will be much richer.
This is why we still call a long-range entangled state with broken symmetry a "spin liquid" and not a SDW, for example. But you're absolutely right that the terminology is confusing and could be improved.
This post imported from StackExchange Physics at 2014-03-09 08:42 (UCT), posted by SE-user Physics Monkey